I've told this story before, I'll tell it again.
One year in uni I was living with another guy and 3 girls in a house. We had a mouse problem and had set up traps to try and cut down/eliminate it.
One day I get home from school and the girls are screaming about a half dead mouse in one of the traps. Apparently it had set off the trap, but the trap hadn't snapped the spine, or cut off the head or whatever and this thing was scurrying back and forth trying to get free with half his body (while the other half looked dead and just lay on the trap).
So I go and get a plastic bag, grab the mouse and trap like I'm picking up dog poop, and then start tying it off and walking outside to stomp on the thing and put it out of it's misery. The girls get upset and say that isn't humane/nice so I ask what they'd rather and they say they'd rather I fill the bag with water and let it drown. So I laughed a bit, said that was silly and that stomping on it was immediate where drowning it was slow and probably painful but they didn't budge, so I filled the bag with water, put it outside in the freezing cold (because this was around January in Canada) and left it until we had a mouse-icle. Then I threw it in the garbage bin.
So in a long winded way I guess I'm saying, I don't really care about killing a mouse and don't see the big deal or need for a 'happy medium'.
The quicker you kill it, the better/nicer you're being as I see it, but clearly people want some sort of magical middle ground where they don't have to kill it but don't have to deal with it ever again.
Unfortunately, I don't think there are any 'mouse' sanctuaries. Maybe you can find a pet store that will hang onto it to feed to reptiles or something, I don't know. Otherwise, IMO, the next time you catch one kill it as fast as you can without risking it's escape. Otherwise, as I see it, you're just setting yourself up for another meeting with the pest in the future.